Stiff Fluid in Accelerated Universes with Torsion

Hindawi Publishing Corporation Journal of Gravity Volume 2013, Article ID 306417, 10 pages http://dx.doi.org/10.1155/2013/306417

Research Article Stiff Fluid in Accelerated Universes with Torsion

Almaz Galiakhmetov

Department of Physics, Donetsk National Technical University, Kirova Street 51, Gorlovka 84646, Ukraine

Correspondence should be addressed to Almaz Galiakhmetov; [email protected]

Received 16 February 2013; Accepted 18 March 2013

Academic Editor: Sergei Odintsov

Copyright © 2013 Almaz Galiakhmetov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The flat Friedmann universes filled by stiff fluid and a nonminimallyupled co material scalar field with polynomial potentials of the fourth degree are considered in the framework of the Einstein-Cartan theory. Exact general solution is obtained for arbitrary positive values of the coupling constant 𝜉. A comparative analysis of the cosmological models with and without stiff fluid is carried out. Some effects of stiff fluid are elucidated. It is shown that singular models with a de Sitter asymptotic and with the power-law 4/3 (𝑡 ) asymptotic at late times are possible. It is found that 𝜉=3/8is a specific value of the coupling constant. It is demonstrated that the bouncing models without the particle horizon and with an accelerated expansion by a de Sitter law of an evolution at late times are admissible.

1. Introduction has grown in connection with the fact that torsion arises naturally in the supergravity [33–35], Kaluza-Klein [36–38], Recent cosmic observations [1–6]favoranisotropicspatially and syperstring [39–41]theories.𝑓(𝑅) gravity with torsion flat Universe, which is at present expanding with acceleration. has been developed [42–45] as one of the simplest extensions The source of this expansion is an unknown substance with of the ECT. In [43] it has been demonstrated that, in 𝑓(𝑅) negative pressure called dark energy (DE). Establishment of gravity, torsion can be a geometric source for the accelerated the origin of DE has become an important problem. Different expansion. The equivalence between a Brans-Dicke theory theoretical models of DE have been put forward (see, e.g., with the Brans-Dicke parameter 𝜔0 =−3/2and the Palatini the reviews [7–10] and references therein). Among these 𝑓(𝑅) models various modifications of general relativity (GR) were gravity with torsion has been demonstrated in [44]. 𝑓(𝑇) considered, the Einstein-Cartan theory (ECT) in particular gravity has been constructed [46–48] as the extension [11–13]. This theory14 [ –17] is an extension of GR to a of the “teleparallel” equivalent of GR [49], which uses the space time with torsion, and it reduces to GR when the Weitzenbock¨ connection that has no curvature but only torsion vanishes. The ECT is the simplest version of the torsion. Poincare´ gauge theory of gravity (PGTG). It should be noted At present, the considerably increased precision of mea- that the ECT contains a nondynamic torsion, because its surements in the modern observational cosmology stipu- gravitational action is proportional to the curvature scalar lated its essential progress. In this connection, the exact of the Riemann-Cartan space-time. In this sense, the ECT is cosmological solutions, which makes it possible to elucidate a degenerate gauge theory [17–20].Thisdrawbackisabsent the detailed picture of an evolution of models, are of great in the PGTG since its gravitational Lagrangian includes interest. It is well known that in GR the exact solutions of the invariants quadratic in the curvature and torsion tensors. Friedmann-Robertson-Walker (FRW) cosmological models Nevertheless the ECT is a viable theory of gravity whose constitute a basis for comparison of theoretical predictions observational predictions are in agreement with the classical with observations. On the other hand, the problem of the testsofGR,anditdifferssignificantlyfromGRonlyatvery cosmological perturbations may be correct, if it is constructed high densities of matter [17, 21, 22]. against a background of the exact solution. The ECT finds applications in cosmology [23–28], par- Other problems of the cosmology, singularities, horizons, ticle theory [19, 29, 30], and the theory of strong interac- and so forth, remain no less acute problems. The main aim of tions [31, 32]. From some time past, the interest to ECT this paper is the investigation of the possibility of the solution 2 Journal of Gravity of some problems of the modern cosmology in the frame- cosmological models with de Sitter asymptotics for work of the ECT. Furthermore, the exact integration of the the late times. ECT equations for the different combination of sources and the comparative analysis of the corresponding cosmological (ii) For 𝜉<0the five types of singular models exist: models are of interest in their own right, since they allows to for the first model the scale factor 𝑎(𝑡) as 𝑡→∞ 𝑝 elucidate the role of the sources of the gravitational field in grows by the law 𝑎∼𝑡 (𝑝<1); for the second cosmology. model 𝑎(𝑡) asymptotically tends to a finite value; other In the framework of the ECT with a nonminimally three models expand from an initial singularity, reach coupled material scalar field that has polynomial potentials themaximum,andthenbegintocontracttoafinal of the fourth degree, we here study isotropic spatially flat singularity. cosmologies in which stiff fluid is taken into account. The motivation to investigate a nonminimally coupled scalar field (iii) There are the specific values of the parameter 𝜉: isbasedontheresultsobtainedwithintheframeworkof 1/6, −1/6, and −3/2. The specific feature of the value the torsionless theories of gravity in connection with the 𝜉=1/6consists in the behavior of the scale factor, inflationary cosmology (see, e.g., [50–54]) and DE model when (𝑎 −min 𝑎 )/𝑎min ≪1: 𝑎≃𝑎min(1 + 𝑦),where 3 2 building [55–57]. On the other hand, the consideration of this 𝑦∼𝑡 for 𝜉=1/6,while𝑦∼𝑡 for 𝜉>1/6.The versionoftheECTincomparisonto𝑓(𝑅) gravity with torsion value 𝜉=−1/6is singled out as only in this case there and 𝑓(𝑇) gravity is motivated by the fact that, in my opinion, exists the singular expanding model, which has the the ECT requires a further study. It is relevant to remark here asymptotic 𝑎|𝑡→+∞ ≃ const. From three types of the that, asit is well known, in the literature there exist a limited recollapsing models with 𝜉<−1/6only for 𝜉=−3/2 number of the exact cosmological solutions in the framework we have the following behavior of the scale factor: −𝛽𝑡 of the ECT. 𝑎|𝑡→+∞ ∼𝑒 , while for others 𝜉,weget The interest in a scalar field potential 𝑉(Φ) in relativis- tic theories of gravity has been aroused by a number of 3 1 𝑎| ∼𝑡(1−√6|𝜉|)/(3−√6|𝜉|) − <𝜉<− , circumstances: its role in cosmology with a time variable 𝑡→+∞ for 2 6 cosmological constant [58] and in quantum cosmology [59]; (1) models with 𝑉(Φ) arise in alternative theories of gravity (1−√6|𝜉|)/(3−√6|𝜉|) 3 𝑎|𝑡→+𝑡 ∼(𝑡0 −𝑡) for 𝜉<− . [60, 61] and supergravity [62]; a scalar potential governs an 0 2 inflation63 [ ] and is actively used in theories of dark matter (DM) and dark energy (DE) [7–10]. It is well known from 𝑉(Φ) > GR that a period of accelerated expansion requires The exact general solutions for spatially flat FRW cos- 0 . In the paper we will consider cosmological models with mologies with a nonminimally coupled scalar field that has 𝑉(Φ) > 0 𝑉(Φ) < 0 and .Thereasonstostudycosmology polynomial potentials of the fourth degree 𝑉(Φ) have been with negative potentials were considered in [64]. obtained in [13]. The main results for the material scalar field A stiff cosmological fluid, with pressure equal to the are as follows. energy density, can be described by a massless scalar field, which is predicted by the string theory. On the other hand, 2 −1/2 (a) For 𝜉>0, Φ <(𝜅𝜉) ,and𝑉(Φ) >,the 0 thestifffluidisanimportantcomponentbecause,atearly solution describes bouncing models of four types with times, it could describe the shear dominated phase of a the accelerated expansion at late times by the laws possible initial anisotropic scenario and is dominating the 𝑎∼𝑒𝐻𝑡 𝑎∼𝑡4/3 Φ2 =(𝜅𝜉)̸ −1/2 𝑉(Φ) < 0 remaining components of the model [65]. Of no little interest and .For , , is the circumstance that this fluid leads to integrability of the the qualitative character of the model evolution does Einstein-Cartan equations. not change. In [66], the flat Friedmann universes filled by radiation, 𝜉<0 Φ 𝑉(Φ) > 0 stiff fluid and a nonminimally coupled ghost scalar field with (b) For ,forall , ,therearetwotypesof polynomial potentials of the fourth degree 𝑉(Φ) have been collapsing models and one type of expanding models. For 𝜉=−3/2there exists a unified model of DM and investigated in the framework of the ECT. Exact particular 𝑉(Φ) < 0 solutionshavebeenobtainedandanalyzed.Theroleof DE.Thecasewith describes the recollapsing sourcesintheevolutionofmodelshasbeenelucidated. models. The exact general solution of an analogous problem for models containing only a nonminimally coupled scalar field The paper is organized as follows. In the next section with an arbitrary coupling constant 𝜉 has been obtained in we present the model and corresponding field equations. In [12]. An analysis of the solutions for the material scalar field Section 3, we obtain exact general solution of the Einstein- 2 −1/2 has shown the following. Cartan equations for 𝜉>0, Φ <(𝜅𝜉) .InSection 3.1 we consider the models with a scalar-torsion field and stiff fluid and discuss some interesting particular cases. In Section 3.2 (i) For 𝜉>0the solution exists for 𝜉≥1/6, we analyze the mixture of stiff fluid with a scalar-torsion 2 −1/2 Φ <(𝜅𝜉) ,where𝜅 is Einstein’s constant only field that has polynomial potentials of the fourth degree. We and describes a countable number of nonsingular summarize our results in Section 4. Journal of Gravity 3

2. Field Equations where 1 The Lagrangian Ł of the model is chosen in the form [13] 𝑇𝑠 =𝛼 {Φ Φ − [Φ Φ,𝑚 +𝜉𝑅({ }) Φ2 −2𝛼𝑉 (Φ)]𝑔 𝑖𝑗 𝑠 ,𝑖 ,𝑗 2 ,𝑚 𝑠 𝑖𝑗 +𝜉[−4𝑆 ∇ +2𝑔 𝑆𝑛∇ −∇∇ +𝑔 ◻ 𝑅 𝛼 (𝑖 𝑗) 𝑖𝑗 𝑛 𝑖 𝑗 𝑖𝑗 =− + ( 𝑠 )[Φ Φ,𝑘 +𝜉𝑅Φ2] −𝑉(Φ) + , Ł ,𝑘 Łfl (2) 2𝜅 2 2 +𝑅𝑖𝑗 ({ }) −Λ𝑖𝑗 ]Φ }, (8) where 𝑅(Γ) is the curvature scalar obtained from the full Γ𝑘 ={𝑘 }+𝑆𝑘 +𝑆𝑘 +𝑆𝑘 { 𝑘 } 𝑇fl =(𝜀 +𝑃 )𝑢𝑢 −𝑃 𝑔 , connection 𝑖𝑗 𝑖𝑗 𝑖𝑗⋅ ⋅𝑖𝑗 ⋅𝑗𝑖; 𝑖𝑗 are the Christoffel 𝑖𝑗 fl fl 𝑖 𝑗 fl 𝑖𝑗 (9) 𝑆𝑘 =Γ𝑘 symbols of the second kind; 𝑖𝑗⋅ [𝑖𝑗] is the torsion tensor; 8 4 𝜅=8𝜋𝐺 𝐺 𝑉(Φ) Λ = 𝑆 𝑆 − 𝑆 𝑆𝑘𝑔 . ,where is the Newtonian constant; is the 𝑖𝑗 3 𝑖 𝑗 3 𝑘 𝑖𝑗 (10) potential of a scalar field;fl Ł is the Lagrangian of stiff fluid; 𝛼 =+1 𝛼 =−1 𝜀 𝑃 𝑠 conforms to the material scalar field; 𝑠 Here fl and fl are the energy density and pressure of stiff corresponds to the ghost scalar field. fluid, respectively; ◻ is the d’Alembertian operator of the 𝑔 −, −, −, + 𝑖 The metric 𝑖𝑘 has the signature ( ); the Riemann Riemannian space; 𝑢𝑖 is the four-velocity (𝑢𝑖𝑢 =1); Ψ= 2 −1 󸀠 and Ricci tensors are defined as 𝜅(𝛼𝑠 −𝜅𝜉Φ ) ,and𝑉 = 𝜕𝑉/𝜕Φ. It is not difficult to verify that the effective scalar-torsion 𝑇𝑠(eff) 𝑚 𝑚 𝑚 𝑚 𝑝 𝑚 𝑝 energy-momentum tensor 𝑖𝑗 , 𝑅𝑖𝑗𝑘⋅ =Γ𝑗𝑘,𝑖 −Γ𝑖𝑘,𝑗 +Γ𝑖𝑝 Γ𝑗𝑘 −Γ𝑗𝑝Γ𝑖𝑘 (3) 𝑠(eff) 𝑠 −1 𝑇𝑖𝑗 =𝑇𝑖𝑗 +𝜅 Λ 𝑖𝑗 , (11)

𝑅 =𝑅𝑖 fl and 𝑗𝑘 𝑖𝑗𝑘⋅ .WeshouldnotethatintheframeworkofECT, and the stiff fluid energy-momentum tensor 𝑇𝑖𝑗 are separately a scalar field nonminimally coupled to gravity gives rise to covariantly conserved since no explicit coupling is assumed torsion, even though the scalar field has zero spin. It follows between stiff fluid and Φ: from (2) that the torsion can interact with a scalar field only 𝑘 ∇𝑗𝑇𝑠(eff) =∇𝑗𝑇fl =0. through its trace: 𝑆𝑖 =𝑆𝑖𝑘⋅ [67]. Hence, the curvature scalar 𝑖𝑗 𝑖𝑗 (12) 𝑗𝑘 𝑅(Γ) = 𝑔 𝑅𝑗𝑘 canbepresentedintheform[67] For spatially flat isotropic and homogeneous models with the metric 8 2 2 2 2 2 2 2 2 𝑅 (Γ) =𝑅({ }) +4∇𝑆𝑘 −( )𝑆 𝑆𝑘, 𝑑𝑠 =𝑎 (𝜂) [−𝑑𝑟 −𝑟 (𝑑𝜃 + sin 𝜃𝑑𝜑 )+𝑑𝜂 ]. (13) 𝑘 3 𝑘 (4) Equations (5)and(7)taketheform

𝑅({}) 𝑎󸀠󸀠 𝑎󸀠2 𝑎󸀠 where is the Riemannian part of the curvature built 2 − = Ψ [2𝜉ΦΦ󸀠󸀠 +2𝜉 ΦΦ󸀠 from the Christoffel symbols; ∇𝑘 is the covariant derivative 𝑎 𝑎2 𝑎 of the Riemannian space. 1 2 One can note that the Lagrangian (2)isacovariant +(− +2𝜉+3𝜉2Φ2Ψ) Φ󸀠 ] (14) generalization of its counterpart in GR, and, for 𝜉=1/6,we 2 obtain the so-called conformal coupling (for 𝑉(Φ) = 0 in the 𝛼 =−1𝜉= 2 torsionless theory). As shown in [67], when 𝑠 , +𝛼𝑠𝑎 Ψ[𝑉(Φ) −𝑃 ], 2 2 fl −1/6, 𝑉(Φ) = −(𝑚 /2)Φ , the scalar field corresponding to 󸀠2 󸀠 𝑎 1 2 𝑎 theLagrangian(2)istheaxionfieldinGR.Incosmologythe =Ψ[( −𝜉2Φ2Ψ) Φ󸀠 +2𝜉 ΦΦ󸀠] axion field is a cold dark matter candidate (see, e.g.,67 [ , 68] 𝑎2 6 𝑎 and references therein). (15) 𝛼 2 Varying the action with the Lagrangian (2)in𝑔𝑖𝑗 , 𝑆𝑘, + 𝑠 𝑎 Ψ[𝑉(Φ) +𝜀 ], and Φ, we obtain the following set of equations for the 3 fl gravitational fields and matter: 𝑎󸀠 𝑎󸀠󸀠 (1 − 6𝜉2Φ2Ψ) (Φ󸀠󸀠 +2 Φ󸀠)+6𝜉 Φ 𝑎 𝑎 (16) 𝑠 fl 2 󸀠 𝛼 2 2 󸀠2 𝐺𝑖𝑗 ({ }) =𝜅(𝑇𝑖𝑗 +𝑇𝑖𝑗 )+Λ𝑖𝑗 , (5) +𝛼𝑎 𝑉 −6 𝑠 𝜉 Ψ ΦΦ =0, 𝑠 𝜅 3 𝑆𝑘 = 𝜉ΨΦΦ󸀠𝑘, 2 (6) where the prime denotes differentiation with respect to 𝜂. Here it is relevant to remark that the use of angular 󸀠 ◻Φ − 𝜉Φ𝑅 (Γ) +𝛼𝑠𝑉 =0, (7) coordinates in metric (13) makes it possible to investigate 4 Journal of Gravity the horizon problem. The passage to the cosmic synchronous where 𝑡 time is fulfilled with the help of the expression (𝐶 +2𝐶 ) 1/4 −1/2 1 fl 𝛽=(𝜅𝜉) ,𝐵=(𝜅𝜉) ,𝑑1 = , ∫ 𝑎(𝜂)𝑑𝜂=∫ 𝑑𝑡. (17) 6𝜉

𝜅 2 +1, for 𝑉 (Φ) >0, For the stiff fluid we have 𝑑2 = 𝐶2𝛽 ,𝑛=±1,𝑚={ −6 3 −1, for 𝑉 (Φ) <0. 𝑃 =𝜀 =𝐶 𝑎 , (18) fl fl fl (24) 𝐶 >0 where fl is a constant. Adding (14)and(15)weget An analysis has shown that, depending on the choice of the gravitational field sources, different types of models are 𝑎󸀠󸀠 𝑎󸀠 2 =Ψ[𝜉ΦΦ󸀠󸀠 +2𝜉 ΦΦ󸀠 + 𝛼 𝑎2𝑉 (Φ) possible. 𝑎 𝑎 3 𝑠 3.1. “Scalar-Torsion Field” + “Stiff Fluid”. Let us consider 1 2 𝛼 +(− +𝜉+𝜉2Φ2Ψ) Φ󸀠 ]+ 𝑠 𝑎2Ψ(𝜀 −3𝑃 ). the models without the scalar field potential 𝑉(Φ).Itis 6 6 fl fl convenient to discuss separately the cases 𝐶1 =0and 𝐶1 =0̸ . (19) 𝐹= The substitution of19 ( )into(16)leadsto 3.1.1. Case (C1=0). From (23), it is possible to express 𝐹(𝑢) 󸀠 explicitly as 𝑎 2 Φ󸀠󸀠 +2 Φ󸀠 −𝜉ΦΨΦ󸀠 +𝛼𝑎2 [𝑉󸀠 +4𝜉ΦΨ𝑉 Φ ] 𝑠 ( ) 󵄨 󵄨 2𝑛/√6𝜉 𝑎 󵄨 𝑢 󵄨 (20) 𝐹=𝐷(󵄨 ( )󵄨) , (25) 1 󵄨tanh 2 󵄨 +𝛼𝜉𝑎2ΦΨ (𝜀 −3𝑃 )=0. 󵄨 󵄨 𝑠 fl fl where 𝐷1 >0is an integration constant. From (20) for stiff fluid and the scalar field potential For 𝑛=+1solution (23) describes bouncing models. The 2 2 minima of the scale factor, 𝑉 (Φ) =𝐶2(𝛼𝑠 −𝜅𝜉Φ ) (21) 󵄨 𝑢 󵄨 −1/√6𝜉 we obtain −1/2 󵄨 min 󵄨 𝑎min =𝛽𝐷1 cosh 𝑢min(󵄨tanh ( )󵄨) , 󵄨 󵄨 󵄨 󵄨 1/2 1/2 󵄨 2 󵄨 󵄨Φ󸀠󵄨 (󵄨𝛼 −𝜅𝜉Φ2󵄨) 𝑎2 (𝜂) = (𝐶 +2𝛼𝜅𝜉ℓ𝐶 Φ2) , (26) 󵄨 󵄨 󵄨 𝑠 󵄨 1 𝑠 fl Φ =𝐵 𝑢 , (22) min tan min where 𝐶1 and 𝐶2 are integration constants; ℓ=sgn(𝛼𝑠 − are defined from 2 𝜅𝜉Φ ). √ 2 It must be noted here that the requirement of renormal- 6𝜉 sinh 𝑢min = cosh 𝑢min. (27) ization of the quantum theory of a scalar field results in the introduction of a nonminimal coupling and the potential in The asymptotic behavior of 𝑎(𝑡),when(𝑎 −min 𝑎 )/𝑎min ≪ the form of (21)[69]. 1,is In this paper, we will restrict our discussion to the 2√1 + 24𝜉 𝐶 𝐷3 material scalar field (𝛼𝑠 =+1).Itfollowsfrom(22)thatthere fl 1 2 𝑎=𝑎min {1 + tanh 𝑢min exist solutions for the following restrictions: 𝛽2 (1 + √1 + 24𝜉) 2 2 𝜉<0,forallΦ and 𝜉>0, Φ =𝐵̸ .Itisnotdifficultto (28) 𝑎(𝑡) 󵄨 󵄨 6/√6𝜉 show that, in terms of physics, reasonable solutions for 󵄨 𝑢min 󵄨 2 2 2 ×(󵄨 ( )󵄨) 𝑡 }. and Φ(𝑡) do not exist for 𝜉<0,forallΦ and 𝜉>0, Φ >𝐵. 󵄨tanh 2 󵄨

3. Exact Solution For 𝑢 ∈ (0, ∞) solution (23) has the following asymptotics: The exact general solution to the Einstein-Cartan equations 2 2 𝑎| ∼ (−𝑡)1/3,Φ| ∼ (−𝑡)−√6𝜉/3, for 𝜉>0, Φ <𝐵can be presented in the form of 𝑡→−∞ 𝑡→−∞ (29) quadratures: 𝐻1𝑡 𝑎|𝑡→+∞ ∼𝑒 ,Φ|𝑡→+∞ ≃𝐵, −1/2 𝑎 (𝑢) =𝛽cosh 𝑢𝐹 ,Φ(𝑢) =𝐵tanh 𝑢, 𝐻 =𝛽−1𝐷3/2(2𝐶 )1/2 𝛽𝑑𝑢 where 1 1 fl . ∫ = ∫ 𝑑𝑡, It should be observed here that we cannot express 𝑎min 𝐹3/2√𝐶 +2𝐶 2𝑢 and 𝑢min in terms of the present values of the cosmographic 1 fltanh (23) parameters 𝐻0,𝑞0, and 𝑗0,sincethreeunknownvalues𝜉, 𝐷1, 𝐹1/2𝑑𝐹 2𝑛𝑑𝑢 and 𝐶 define the Hubble parameter 𝐻1. ∫ = ∫ , fl It is easy to see that Hubble’s parameter 𝐻1 can take √𝑑 𝐹3 +𝑚𝑑 𝑢√𝐶 +2𝐶 2𝑢 𝜉≪1 1 2 cosh 1 fltanh large values if . Since the models are considered in Journal of Gravity 5 the framework of a classical theory of gravity, they will be It must be noted that, for √6𝜉 = 3,thesolutionmaybe 𝜀<𝜀 𝜀 physically admissible provided pl,where is the total written in a parametric form energy density of the scalar-torsion field and stiff fluid and 𝜀 pl is the Planck energy density. Consequently, the following 𝜉 𝜉≪1 1/3 restriction on is valid for : −1/2 −1/3√ 2 √ 2 𝑎 (𝑡) =𝛽𝐷1 𝑥 1+𝑥 (1 + 1+𝑥 ) , 𝜉 > 36𝜅−3𝜀−1𝐷6𝐶2 . pl 1 fl (30) 𝑛 𝑥 Φ (𝑡) =𝐵 1 , It is interesting to observe that the models (29)arefree √1+𝑥2 from the particle horizon; they are nonsingular and admit the 9𝐻2𝑥6 (36) late-time accelerating expansion. 𝑆2 (𝑡) = 1 , 2 For 𝑢∈(−∞,0)themodelswiththereverseasymptotics 4(1 + 𝑥2)2(1 + √1+𝑥2) exist: 󵄨 󵄨 −1 −𝐻 𝑡 󵄨 √ 2󵄨 √ 2 1 ln 󵄨𝑥+ 1+𝑥 󵄨 −𝑥 (1 + 1+𝑥 )=𝑛1𝐻1𝑡, 𝑎|𝑡→−∞ ∼𝑒 ,Φ|𝑡→−∞ ≃−𝐵, 󵄨 󵄨 (31) 1/3 −√6𝜉/3 𝑎|𝑡→+∞ ∼𝑡 ,Φ|𝑡→+∞ ∼−𝑡 . where 𝑥 ∈ (0, ∞), 𝑛1 =+1for 𝑢 ∈ (0, ∞) and 𝑛1 =−1for It follows from (31)thatatlatetimesthescalarfield 𝑢∈(−∞,0). disappears with time, and the behavior of the scale factor For 𝑛=−1, 𝑢∈(0,∞),solution(23) corresponds to the corresponds to the decelerated expansion. singular expanding models with de Sitter asymptotics for the It is easy to verify that models (29)and(31) are regular late times: duetotheviolationofthestrongenergyconditionwitha scalar-torsion field. It is not difficult to show that, for minima of the scale factor and for the de Sitter-like asymptotics, the 1/3 √6𝜉/3 𝑎|𝑡→−𝑡 ∼(𝑡0 +𝑡) ,Φ|𝑡→−𝑡 ∼(𝑡0 +𝑡) , contribution of the scalar-torsion field dominates, while for 0 0 1/3 (37) 𝑎∼𝑡 the contribution of the stiff fluid dominates. 𝐻 𝑡 𝑎| ∼𝑒 1 ,Φ| ≃𝐵, As pointed out in [70, 71], since the current torsion is 𝑡→+∞ 𝑡→+∞ extremely small, only those solutions should be considered as physical solutions towards current epoch, in which torsion disappears with time. In this connection, it should be noted while for 𝑢∈(−∞,0),thissolutionconformstothemodels 2 𝑘 thatthesquareofthetraceoftorsion𝑆 =𝑆𝑘𝑆 has the with the reverse asymptotics, which describe the collapsing following asymptotics: models. It is easy to verify that for 𝑡→−𝑡0 the contribution of 2 (1) stiff fluid dominated era thestifffluiddominates.Thebehaviorof𝑆 for 𝑡→+∞ is described by the formula (33). It is interesting to observe 𝑆2| ∼ (∓𝑡)−2(3+2√6𝜉)/3 󳨀→ 0, 𝑡→∓∞ (32) that the square of the trace of torsion has the following asymptotics for 𝑡→−𝑡0: (2) de Sitter regime

2 9 2 𝑆 |𝑡→∓∞ ≃( )𝐻 = const. (33) 3 4 1 { →∞, √6𝜉 < , { for 2 { { It follows from (32)thatforthestifffluiddominatedera 2 2(2√6𝜉−3)/3 3 𝑆 |𝑡→−𝑡 ∼(𝑡0 +𝑡) = {= const, for √6𝜉 = , the torsion disappears with time. The formula (33)showsthat 0 { 2 { for the de Sitter regime the torsion asymptotically tends to a { 3 𝐻 =𝐻 →0, √6𝜉 > . finitevalue.Forthecurrentepoch,providedthat 1 0, { for 2 𝐻 where 0 is the present-day value of the Hubble’s parameter, (38) we have the following in natural units (ℏ=𝑐=1):

2 −84 2 𝑆 |𝑡→+∞ ∼10 𝐺𝑒𝑉 . (34) That is, there is the specific value of the coupling constant: That is, the current torsion is very small. 𝜉=3/8. 𝑎| ∼ For minima of the scale factor the square of the trace of Thus, for the Big Bang singularity of the form 𝑡→−𝑡0 1/3 torsion is (𝑡0 +𝑡) : 𝑎→∞, 𝜀→∞, 𝑃→∞,where𝑃 is the total 󵄨 󵄨 6/√6𝜉 pressure of the scalar-torsion field and stiff fluid, the torsion 2 9 2 4 󵄨 𝑢min 󵄨 𝑆 =( )𝐻 𝑢 (󵄨 ( )󵄨) = . √6𝜉 < 3/2 √6𝜉 = 3/2 4 1 tanh min 󵄨tanh 2 󵄨 const canbesingularfor , constant for ,or tend to zero for √6𝜉 > 3/2.Thesolutionsforthethreetypes (35) 2 of 𝑆 behaviorcanbepresentedinaparametricform(Φ(𝑢) = That is, the torsion is finite. 𝐵 tanh 𝑢): 6 Journal of Gravity

(1) √6𝜉 = 3/4 3𝑛/√6𝜉 , where 𝑊=𝐷2(| tanh(𝑢/2)|) ; 𝐷2 >0is an integration 𝑢 constant. −1/2 4/3 𝑉(Φ) < 0 𝑎 (𝑢) =𝛽𝐷1 cosh 𝑢 tanh ( ), An analysis has shown that this solution with , 2 2 for all 𝑛 and 𝑉(Φ) >, 0 𝑛=−1,and𝐷2 >𝑑2 for 𝑢∈ 9 𝑢 (0, ∞) 𝑆2 (𝑢) =( )𝐻2 4𝑢 −8 ( ), describes singular expanding models with the de Sitter 4 1 tanh tanh 2 asymptotic at late times: −1 (39) 2 𝑢 2 𝑢 1/3 √6𝜉/3 2 2(2√6𝜉−3)/3 (5 + ( )) (2 ( )) 𝑎|𝑡→0 ∼𝑡 ,Φ|𝑡→0 ∼𝑡 ,𝑆|𝑡→0 ∼𝑡 , tanh 2 cosh 2 𝐻 𝑡 2 9 2 𝑢 𝑎| ∼𝑒 3 ,Φ|≃𝐵, 𝑆| ≃( )𝐻 , +4 ( ( )) = 2𝐻 (𝑡 +𝑡), 𝑡→∞ 𝑡→∞ 𝑡→∞ 4 3 ln cosh 2 1 0 (44) −1 where 𝑡0 = 5(4𝐻1) . 1/2 −1 2 where 𝐻3 =(6𝜉) (2𝛽𝐷2) (𝐷2 −𝑚𝑑2).Notethatfor𝑢∈ (−∞, 0) (2) √6𝜉 = 3/2 the reverse asymptotic behavior takes place. , 2 √ For 𝑉(Φ) <, 0 𝑛=+1, 𝐷2 =𝑑2,and 6𝜉 = 3,thesolution 𝑢 𝑎 (𝑢) =𝛽𝐷−1/2 𝑢 2/3 ( ), can be expressed in elementary functions: 1 cosh tanh 2 󵄨 󵄨 1/3 𝑎 (𝑡) =𝐴1 cosh 𝛾𝑡(󵄨tanh 𝛾𝑡󵄨) ,Φ(𝑡) =𝐵tanh 𝛾𝑡, 2 9 2 4 −4 𝑢 𝑆 (𝑢) =( )𝐻1 tanh 𝑢 tanh ( ), (40) 4 2 2 9 2 2 𝑆 (𝑡) =( )𝐻3 tanh 𝛾𝑡, 𝑢 𝑢 4 −2 ( )+4 ( ( )) = 2𝐻 (𝑡 +𝑡), cosh 2 ln cosh 2 1 0 (45) −1/3 1/6 −1 −1 where 𝐴1 = 𝛽(3𝐷2) (2𝐶 ) , 𝛾=3𝛽 𝐷2. where 𝑡0 =(2𝐻1) . fl It should be pointed out that, for 𝑡 ∈ (0, ∞),wehave 𝑡∈(−∞,0) (3) √6𝜉 = 3, the expanding models, while, for ,wegetthe collapsing models. 󵄨 󵄨 1/3 𝑉(Φ) > 0 𝑛=−1𝐷2 =𝑑 𝑢 ∈ (0, ∞) −1/2 󵄨 𝑢 󵄨 For , , 2 2,and ,this 𝑎 (𝑢) =𝛽𝐷 𝑢(󵄨 ( )󵄨) , 1 cosh 󵄨tanh 2 󵄨 solution corresponds to the singular expanding models with the power-law asymptotic at late times: 9 𝑢 𝑆2 (𝑢) =( )𝐻2 4𝑢 −2 ( ), (41) 4 1 tanh tanh 2 1/3 √6𝜉/3 𝑎|𝑡→0 ∼𝑡 ,Φ|𝑡→0 ∼𝑡 , 𝑢 √ 𝑢−tanh ( )=2𝐻1 (𝑡0 +𝑡), 2 2(2 6𝜉−3)/3 2 𝑆 |𝑡→0 ∼𝑡 , (46) 𝑡 =0 4/3 where 0 . 𝑎|𝑡→+∞ ∼𝑡 ,Φ|𝑡→+∞ ≃𝐵, 𝑆2| ∼𝑡−2 󳨀→ 0. 3.1.2. Case (C1 ≠ 0). An analysis has shown that for 𝐶1 >0 𝑡→+∞ the solution exists only for 𝑛=−1and describes nonsingular It is easy to see that the case for 𝑢∈(−∞,0)conforms to the expanding models with the asymptotics: models with the reverse asymptotics. 𝐶 +2𝐶 It should be observed here that the accelerated expansion 𝑎| ≃𝑎 (1 + 𝛽−1𝐷3/2√ 1 fl 𝑡) , of models with the power-law asymptotic at late stages of the 𝑡→0 01 1 6𝜉 (42) cosmological evolution has its origin in “Scalar-torsion field” +“𝑉(Φ).” 𝐻2𝑡 Φ|𝑡→0 ∼𝑡, 𝑎|𝑡→+∞ ∼𝑒 ,Φ|𝑡→+∞ ≃𝐵, For √6𝜉 = 3,thesolutioncanbepresentedinelementary

−1/2 −1 3/2 1/2 functions: 𝑎01 =𝑎(𝑡=0)=𝛽𝐷 𝐻2 =𝛽 𝐷 (𝐶1 +2𝐶 ) where 1 , 1 fl . 1/2 𝛾𝐵𝑡 𝐶 <0 1/3 2 2 It is not difficult to show that for 1 ,wehavethesame 𝑎 (𝑡) =𝐴2(|𝑡|) (1 + 𝛾 𝑡 ) ,Φ(𝑡) = , 2 2 qualitative picture of the evolution of models as for 𝐶1 >0. √1+𝛾 𝑡

𝑉(Φ 9𝛾4𝑡2 3.2. “Scalar-Torsion Field” + “ )” + “Stiff Fluid”. Let us 𝑆2 (𝑡) = , consider the models for which all sources of gravitational field 4(1 + 𝛾2𝑡2)2 from the Lagrangian (2)aretakenintoaccount. (47) 𝐹(𝑢) 𝐴 =𝛽2/3(2𝐶 )1/6 𝑡→0 3.2.1. Case (C1=0). From (23) we find the expression for , where 2 fl .Itfollowsfrom(47)thatfor thesquareofthetraceoftorsionhasthefollowingbehavior: 3/2 2 1/2 −1 2 2 𝐹 =(𝑊 −𝑚𝑑2) (2𝑑1 𝑊) , (43) 𝑆 |𝑡→0 ∼𝑡 →0. Journal of Gravity 7

√ 1/2 2 The analytic solution in elementary functions for 6𝜉 = where 𝜒1 =(𝑔1 −1)[(𝑔1 +1)cosh 𝐷−(𝑔1 −1)](6𝜉) +11𝑔1 − 3/2, 2𝑔1 −1. Φ 𝑆2 1/3 2/3 For (51)and(52), the expressions for and have the 𝑎 (𝑡) =𝐴3{|𝑡| (1 + 𝛾 |𝑡|)} (1 + 2𝛾 |𝑡|) , form

1/2 −1 2 Φ (𝑡) =2𝐵{𝛾|𝑡| (1 + 𝛾 |𝑡|)} (1 + 2𝛾 |𝑡|) , (48) 27𝜉𝑑 (𝑔 −1) Φ (𝑡=0) =𝐵 𝐷,2 𝑆 (𝑡=0) = 2 1 . tanh 2 8𝑔1𝛽 𝑆2 (𝑡) = 54𝜉𝛽−2𝐷2(1 + 2𝛾 |𝑡|)−2, 2 (53) 𝐴 =𝛽2/3(8𝐶 )1/6 where 3 fl , corresponds to the following It is easy to see that for 𝑔1 >1the solution describes the 2 2 −2 2 behavior of 𝑆 for 𝑡=0: 𝑆 = 54𝜉𝛽 𝐷2 = const. expanding nonsingular models. Here it is relevant to remark Formulae (47)and(48) show that the analytic represen- that the accelerated expansion both at early stages and at late tation of the solution depends on the behavior of the square stages of the cosmological evolution is characteristic for case of the trace of torsion at early times. (b). The solution in a parametric form for √6𝜉 = 3/4, It is not difficult to verify that for 𝑢 ∈ (−∞, −𝐷] we get the models with the reverse asymptotics. 1/6 2𝐶 𝑢 For 𝑉(Φ) >, 0 𝑛=+1,and𝑔1 =1,solution(23) describes 𝑎 (𝑢) =2𝛽( fl ) 𝑢 4/3 ( ) cosh tanh bouncing models with the asymptotics 9𝑑2 2 −𝐻 𝑡 𝑢 −1/3 𝑎| ∼𝑒 4 ,Φ| ≃𝐵 𝐷, ×(1− 8 ( )) , 𝑡→−∞ 𝑡→−∞ tanh tanh 2 2 6𝐻4𝑡 𝑆 |𝑡→−∞ ∼𝑒 󳨀→ 0, 1/2 2 81𝑑2 2 4 −8 𝑆 (𝑢) =( )( ) tanh 𝑢 tanh 𝐻3𝑡 (54) 64 3𝜅 (49) 𝑎|𝑡→+∞ ∼𝑒 ,Φ|𝑡→+∞ ≃𝐵, 2 9 𝑢 8 𝑢 2 2 ×( )(1− ( )) , 𝑆 |𝑡→+∞ ≃( )𝐻 = const, 2 tanh 2 4 3 Φ (𝑢) =𝐵 𝑢, −1 tanh where 𝐻4 = 𝜆(𝛽 cosh 𝐷) . The minima of the scale factor, 2 𝑢 2 𝑢 cosh ( )−arctan (tanh ( )) = 1 + 2𝛾 |𝑡| , 2 2 4𝑔 𝐶 1/6 −1/3 2 fl 𝑎min =𝛽(𝑔2 −1) ( ) cosh 𝑢min, 2 3𝜉𝑑 conforms to the following behavior of 𝑆 for 𝑡→0: 2 (55) 2 𝑆 |𝑡→0 →∞. Φmin =𝐵tanh 𝑢min, For 𝑉(Φ) >, 0 𝑛=+1,solution(23) exists for 𝑢∈ (−∞, −𝐷] 𝑢 ∈ [𝐷, +∞) 𝐷 and ,where is a constant. With 𝑔 =(| (𝑢 /2)/ (𝐷/2)|)6/√6𝜉 >1 𝑢 ∈ [𝐷, +∞) provided that where 2 tanh min tanh ,are defined from 󵄨 󵄨 3/√6𝜉 󵄨 𝐷 󵄨 2 1/2 2 𝜆=𝐷(󵄨 ( )󵄨) ,𝜆=𝑔𝑑 ,𝑔>1, (𝑔2 −1)(6𝜉) sinh 𝑢min =(𝑔2 +1)cosh 𝑢min. (56) 2 󵄨tanh 2 󵄨 1 2 1 (50) The asymptotic behavior of 𝑎(𝑡),when(𝑎 −min 𝑎 )/𝑎min ≪ 1,is the evolution of models for 𝑡→∞is identical to the models (44), while for 𝑡→0we have 2 3 −2 2 𝑎=𝑎min [1 + 𝑑2𝜒2(8𝑔2𝛽 cosh 𝑢min) 𝑡 ], (57) 1/2 2 (a) (𝑔1 − 1)(6𝜉) sinh 𝐷>(𝑔1 +1)cosh 𝐷, where 𝑎| ≃𝑎 𝑡→0 02 2 1/2 2 𝜒2 = [12𝑔2 − (6𝜉) (𝑔2 +1) ] cosh 𝑢min (𝑔 −1)(6𝜉)1/2 2𝐷−(𝑔 +1) 𝐷 ×[1+ 1 sinh 1 cosh 𝑑1/2𝑡] , 1/2 2 1/2 2 2 + (6𝜉) (𝑔2 −1), (58) 2𝛽𝑔1 cosh 𝐷 (51) 2 1/2 2 12𝑔2 > (6𝜉) (𝑔2 +1) . −1/3 1/6 𝑎 =𝑎(𝑡=0)=𝛽(𝑔 −1) (8𝑔 𝐶 /6𝜉𝑑 ) 𝐷 2 where 02 1 1 fl 2 cosh , The value of 𝑆 for minima of the scale factor is

1/2 2 2 −1 (b) (𝑔1 − 1)(6𝜉) sinh 𝐷=(𝑔1 +1)cosh 𝐷, 2 2 −2 𝑆min =9𝑑2(𝑔2 +1) (16𝑔2𝛽 ) cosh 𝑢min. (59) 2 2 −1 2 𝑎|𝑡→0 ≃𝑎02 [1 + 𝑑2𝜒1(8𝑔1𝛽 cosh 𝐷) 𝑡 ], (52) That is, the torsion is finite. 8 Journal of Gravity

3.2.2. Case (C1 ≠ 0). An analysis has shown that with 𝐶1 =0̸ at late times corresponds to ΛCDM model, while the behavior 4/3 for all 𝑉(Φ) only the nonsingular expanding models of types like 𝑎∼𝑡 at late times conforms to a quintessence model (42)arepossible. of DE [10, 72].

4. Conclusion Acknowledgments In this paper, exact general solution for spatially flat isotropic The author is grateful to the unknown reviewers for the useful and homogeneous cosmological models in ECT with stiff remarks. fluid and a nonminimally coupled material scalar field with apotential𝑉(Φ) has been obtained and analyzed for an arbitrary positive coupling constant 𝜉. References In order to elucidate the role of the stiff fluid in the [1] A. G. Riess, L.-G. Sirolger, J. Tonry et al., “Type Ia supernova Einstein-Cartan cosmology, we should carry out a com- discoveries at 𝑧>1from the hubble space telescope: evidence parative analysis of the corresponding cosmological models for past deceleration and constraints on dark energy evolution,” obtained in this paper and those in [12] for “Scalar-torsion The Astrophysical Journal,vol.607,article665,2004. 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